Linear-to-circular polarizers using cascaded sheet impedances and cascaded waveplates

ABSTRACT

An ultra-wideband linear-to-circular polarizer is disclosed. In accordance with embodiments of the invention, the polarizer includes a plurality of cascaded waveplates having biaxial permittivity or cascaded anisotropic sheet impedances. Each waveplate/sheet has a principal axis rotated at different angles relative to an adjacent waveplate/sheet about a z-axis of a 3-dimensional x, y, z coordinate system. Each waveplate is composed of a unit cell of an artificial anisotropic dielectric. Each sheet impedance is composed of an anisotropic metallic pattern. The polarizer further includes impedance matching layers disposed adjacent the cascaded waveplates/sheets.

CROSS-REFERENCE TO RELATED APPLICATION(S)

The present application is a continuation of U.S. application Ser. No.16/181,624 filed Nov. 6, 2018, which claims the benefit of U.S.Provisional Application No. 62/594,804, filed Dec. 5, 2017, the contentsof which are hereby incorporated by reference.

STATEMENT OF GOVERNMENT INTEREST

The invention described herein may be manufactured, used, and licensedby or for the Government of the United States for all governmentalpurposes without the payment of any royalty.

BACKGROUND

Linear-to-circular polarizers convert an incident, linearly polarizedplane wave into a transmitted, circularly polarized wave.Linear-to-circular polarizers are commonly utilized from microwave tooptical frequencies for a myriad of applications. Many of theseapplications also demand wide operating bandwidths and wide angles ofincidence. However, conventional linear-to-circular polarizers only workperfectly at a single frequency making them inherently narrowband.

At THz frequencies and higher, wideband linear-to-circular polarizersare typically realized by cascading multiple birefringent waveplateswith rotated principal axes. Polarizers utilizing cascaded waveplatescan realize multiple octaves of bandwidth. At these higher frequencies,the geometry can afford to be many wavelengths in thickness while stillmaintaining a low profile since the wavelength is short. A disadvantageinherent in these designs is that they do not typically work well atwide angles of incidence since the optical thickness of the plate is afunction of the angle of incidence.

At microwave frequencies, the most common linear-to-circular polarizersutilize cascaded patterned metallic sheets (i.e., sheet impedances) withsubwavelength overall thicknesses. The bandwidth of microwavelinear-to-circular polarizers are typically less than 40%. In someexamples, the bandwidth has been increased up to an octave usingmeanderline metallic patterns printed on dielectric substrates. However,these meanderline polarizers do not typically work well at wide anglesof incidence when their bandwidth is large.

Conventional waveplates composed of uniaxial dielectrics (i.e.,ε_(xx)=ε_(zz)≠ε_(yy)) only operate at a single frequency. It has beenknown since the 1950s that the bandwidth can be significantly extendedby cascading waveplates with different thicknesses and relativeorientations to develop so-called achromatic waveplates. Thesewaveplates are commercially available at optical frequencies withbandwidths of over 4:1. While this design approach has been scaled downfrom optical frequencies to THz and mm-waves, as the wavelength isincreased further, the required thickness of naturally occurringcrystals becomes prohibitive due to the notable, weight, size, and loss.

In view of the above, it would be advantageous to provide alinear-to-circular polarizer that provides improved wide-bandperformance.

SUMMARY OF THE DISCLOSURE

There is provided a linear-to-circular polarizer that includes aplurality of cascaded waveplates having biaxial permittivity. Eachwaveplate has a principal axis rotated at different angles relative toan adjacent waveplate about a z-axis of a 3-dimensional x, y, zcoordinate system. Impedance matching layers are disposed adjacent thecascaded waveplates.

In accordance with a further embodiment of the invention, a firstassembly of impedance matching layers is disposed adjacent a firstwaveplate of the cascaded waveplates, and a second assembly of impedancematching layers is disposed adjacent a second waveplate of the cascadedwaveplates. In an example embodiment, the plurality of cascadedwaveplates includes four waveplate assemblies, where each assembly isrotated at a different angle relative to an adjacent waveplate assembly.

In accordance with another embodiment of the invention, a firstwaveplate is rotated at a first angle relative about the z-axis, asecond waveplate is rotated at a second angle about the z-axis, a thirdwaveplate is rotated at a third angle about the z-axis, and a fourthwaveplate is rotated at a fourth angle about the z-axis, where theselection of the first, second and third angles is based on operatingwavelengths of the polarizer.

In accordance with yet another embodiment of the invention, eachwaveplate has a respective length with respect to the z-axis differentfrom a length of an adjacent waveplate.

In accordance with still a further embodiment of the invention, theimpedance matching layers have a biaxial permittivity.

In accordance with yet another embodiment of the invention, theimpedance matching layers include a first assembly of impedance matchinglayers and a second assembly of impedance matching layers, where each ofthe first and second assemblies of impedance matching layers include afirst section having a first permittivity ∈₁, a second section having asecond permittivity ∈₂ greater than the first permittivity, and a thirdsection having a third permittivity ∈₃ greater than the secondpermittivity.

In accordance with a further embodiment of the invention, where in eachof the first and second assemblies of impedance matching layers, thefirst section has a first thickness, the second section has a secondthickness, and the third section has a third thickness less than thefirst thickness and greater than the second thickness.

In accordance with still another embodiment of the invention, eachassembly of impedance matching layers includes a plurality of differentsubstrates.

In accordance with yet another embodiment of the invention, eachwaveplate includes a unit cell of an artificial anisotropic dielectric.

In accordance with still a further embodiment, each unit cell includes asubstrate patterned with a copper patch.

In accordance with another embodiment, there is provided alinear-to-circular polarizer that includes a plurality of cascadedanisotropic sheets. Each sheet has a principal axis rotated at differentangles relative to an adjacent sheet about a z-axis of a 3-dimensionalx, y, z coordinate system. Impedance matching layers, as describedabove, are disposed adjacent the cascaded sheets.

In accordance with yet another embodiment, the cascaded sheets includeanisotropic metallic patterns.

In accordance with still another embodiment, the anisotropic metallicpatterns have meanderline and metallic patch geometries.

In accordance with yet another embodiment, a three dimensional (3D)printed dielectric grating is embedded between the impedance matchinglayers.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings provide visual representations which will beused to more fully describe various representative embodiments and canbe used by those skilled in the art to better understand therepresentative embodiments disclosed and their inherent advantages. Thedrawings are not necessarily to scale, emphasis instead being placedupon illustrating the principles of the devices, systems, and methodsdescribed herein. In these drawings, like reference numerals mayidentify corresponding elements.

FIGS. 1A and 1B are high-level schematic diagrams of an ultra-widebandlinear-to-circular polarizer in accordance with an embodiment of theinvention;

FIG. 2 is a graphical representation of the simulated response when thepermittivities of a pair of waveplates are increased and the polarizeris illuminated at an angle of incidence of 45° in the E- and H-planes;

FIG. 3 is an isometric schematic of a unit cell of the cascaded,anisotropic waveplates in accordance with an embodiment of theinvention;

FIGS. 4A, 4B and 4C, are depictions of the simulated transmissioncoefficient and axial ratio for a polarizer in accordance with anembodiment of the invention when illuminated with an x-polarized planewave for different angles of incidence, where FIG. 4A is normalincidence, FIG. 4B is 45 degrees from normal, and FIG. 4C is 60 degreesfrom normal;

FIGS. 5A, 5B, 5C and 5D are illustrations of a section of an exemplarycascaded sheet impedance polarizer in accordance with an embodiment ofthe invention;

FIGS. 6A, 6B and 6C are graphical representations of simulatedperformance of the polarizer shown in FIGS. 5A, 5B, 5C and 5D;

FIGS. 7A, 7B and 7C are illustrations of a cascaded waveplate polarizerfabricated by stacking together chemically etched printed-circuit-boardsin accordance with an embodiment of the invention;

FIG. 8 is a graphical representation of the cascaded waveplatepolarizer's measured and simulated transmission coefficient (T_(Rx)),and axial ratio (AR) at normal incidence;

FIGS. 9A and 9B are illustrations of a cascaded sheet impedancepolarizer in accordance with an embodiment of the invention fabricatedusing standard PCB processing techniques;

FIGS. 10A, 10B and 10C are graphs of the measured and simulatedperformance of the cascaded sheet impedance polarizer for various anglesof incidence, where FIG. 10A is normal incidence, FIG. 10B is 45 degreesfrom normal, and FIG. 10C is 60 degrees from normal;

FIG. 11 depicts an embodiment of a polarizer in accordance with anembodiment of the invention having an additional anisotropic layer forimproved performance; and

FIGS. 12A and 12B are graphs illustrating the measured performance ofthe polarizer of FIG. 11 at a normal angle of incidence and 60° scanangles.

DETAILED DESCRIPTION

Specific embodiments of the invention will now be described in detailwith reference to the accompanying figures. While this invention issusceptible of being embodied in many different forms, there is shown inthe drawings and will herein be described in detail specificembodiments, with the understanding that the present disclosure is to beconsidered as an example of the principles of the invention and notintended to limit the invention to the specific embodiments shown anddescribed. In the description below, like reference numerals may be usedto describe the same, similar or corresponding parts in the severalviews of the drawings.

All documents mentioned herein are hereby incorporated by reference intheir entirety. References to items in the singular should be understoodto include items in the plural, and vice versa, unless explicitly statedotherwise or clear from the text.

For simplicity and clarity of illustration, reference numerals may berepeated among the figures to indicate corresponding or analogouselements. Numerous details are set forth to provide an understanding ofthe embodiments described herein. The embodiments may be practicedwithout these details. In other instances, well-known methods,procedures, and components have not been described in detail to avoidobscuring the embodiments described. The description is not to beconsidered as limited to the scope of the embodiments described herein.

There is provided a linear-to-circular polarizer that includes aplurality of cascaded waveplates having biaxial permittivity. Eachwaveplate has a principal axis rotated at different angles relative toan adjacent waveplate about a z-axis of a 3-dimensional x, y, zcoordinate system. Impedance matching layers are disposed adjacent thecascaded waveplates.

In accordance with a further embodiment of the invention, a firstassembly of impedance matching layers is disposed adjacent a firstwaveplate of the cascaded waveplates, and a second assembly of impedancematching layers is disposed adjacent a second waveplate of the cascadedwaveplates. In an example embodiment, the plurality of cascadedwaveplates includes four waveplate assemblies, where each assembly isrotated at a different angle relative to an adjacent waveplate assembly.

In accordance with another embodiment of the invention, a firstwaveplate is rotated at a first angle relative about the z-axis, asecond waveplate is rotated at a second angle about the z-axis, a thirdwaveplate is rotated at a third angle about the z-axis, and a fourthwaveplate is rotated at a fourth angle about the z-axis, where theselection of the first, second and third angles is based on operatingwavelengths of the polarizer.

In accordance with yet another embodiment of the invention, eachwaveplate has a respective length with respect to the z-axis differentfrom a length of an adjacent waveplate.

In accordance with still a further embodiment of the invention, theimpedance matching layers have a biaxial permittivity.

In accordance with yet another embodiment of the invention, theimpedance matching layers include a first assembly of impedance matchinglayers and a second assembly of impedance matching layers, where each ofthe first and second assemblies of impedance matching layers include afirst section having a first permittivity ∈₁, a second section having asecond permittivity ∈₂ greater than the first permittivity, and a thirdsection having a third permittivity ∈₃ greater than the secondpermittivity.

In accordance with a further embodiment of the invention, where in eachof the first and second assemblies of impedance matching layers, thefirst section has a first thickness, the second section has a secondthickness, and the third section has a third thickness less than thefirst thickness and greater than the second thickness.

In accordance with still another embodiment of the invention, eachassembly of impedance matching layers includes a plurality of differentsubstrates.

In accordance with yet another embodiment of the invention, eachwaveplate includes a unit cell of an artificial anisotropic dielectric.

In accordance with still a further embodiment, each unit cell includes asubstrate patterned with a copper patch.

In accordance with another embodiment, there is provided alinear-to-circular polarizer that includes a plurality of cascadedanisotropic sheets. Each sheet has a principal axis rotated at differentangles relative to an adjacent sheet about a z-axis of a 3-dimensionalx, y, z coordinate system. Impedance matching layers, as describedabove, are disposed adjacent the cascaded sheets.

In accordance with yet another embodiment, the cascaded sheets includeanisotropic metallic patterns.

In accordance with still another embodiment, the anisotropic metallicpatterns have meanderline and metallic patch geometries.

In particular, for an arbitrary structure illuminated with a normallyincident plane wave, the linearly polarized transmission matrix(T^(LIN)) of the structure relates the incident electric field E_(i) tothe transmitted electric field E_(t):

${\begin{pmatrix}E_{t}^{x} \\E_{t}^{y}\end{pmatrix} = {{T^{LIN}\begin{pmatrix}E_{i}^{x} \\E_{i}^{y}\end{pmatrix}} = {{e^{{- j}\; \delta}\begin{pmatrix}T_{xx} & T_{xy} \\T_{yx} & T_{yy}\end{pmatrix}}\begin{pmatrix}E_{i}^{x} \\E_{i}^{y}\end{pmatrix}}}},$

where δ represents a constant phase shift. An ideal linear-to-circularpolarizer converts an incident x-polarization to a transmittedright-hand circular polarization. This may be represented byT_(xx)=1/√{square root over (2)} and T_(yx)=−j/√{square root over (2)}.It is convenient to characterize the performance of a linear-to-circularpolarizer by considering the linear-to-circular transmission matrix(T^(CP)), which may be defined as:

${\begin{pmatrix}E_{t}^{R} \\E_{t}^{L}\end{pmatrix} = {{T^{CP}\begin{pmatrix}E_{i}^{x} \\E_{i}^{y}\end{pmatrix}} = {\begin{pmatrix}T_{Rx} & T_{Ry} \\T_{Lx} & T_{Ly}\end{pmatrix}\begin{pmatrix}E_{i}^{x} \\E_{i}^{y}\end{pmatrix}}}},$

where R and L denote transmission into right- and left-handed circularpolarizations, respectively. Ideally, T_(Rx)=1 and T_(Lx)=0. Thepolarization purity of the transmitted wave is often expressed in termsof the axial ratio (AR), which can be related to the linear-to-circulartransmission matrix by:

${AR} = {\frac{{{T_{Rx}/T_{Lx}}} + 1}{{{T_{Rx}/T_{Lx}}} - 1}.}$

A y-polarized wave is not considered in this description.

The polarizers described herein are reported at different angles ofincidence, where the E and H planes are defined relative to the plane ofthe incident wave. In this regard, the E-plane corresponds to the ϕ=0°plane and the H-plane is the ϕ=90° plane. It should also be noted thatthe term T_(Rx) characterizes the transmission of both obliquelyincident waves and normally incident waves.

FIGS. 1A and 1B are high-level schematic diagrams of alinear-to-circular polarizer 100 in accordance with an embodiment of theinvention. The linear-to-circular polarizer 100 includes a plurality ofcascaded waveplates 102 ₁ . . . 102 _(N) (where “N” is any suitablenumber) (depicted with four waveplates in the example embodiment)disposed relative to a 3-dimensional x, y and z coordinate system. Thecascaded waveplates 102 ₁ . . . 102 _(N) exhibit biaxial permittivity.Each waveplate has a principal axis rotated at different angles relativeto an adjacent waveplate about the z-axis, and each waveplate can beprovided with a respective length with respect to the z-axis differentfrom a length of an adjacent waveplate. The polarizer 100 furtherincludes impedance matching layers 104 ₁, 104 ₂ that are disposedadjacent, or substantially adjacent, the cascaded waveplates 102 ₁ . . .102 _(N). In the example shown, the overall length of the assembly isapproximately 28.4 mm corresponding to wavelengths of approximately6.6λ_(h). This is merely illustrative, as the configuration is dependentupon the desired operating conditions as explained further below.

For a single waveplate polarizer, ignoring reflection losses andabsorption, the transmission matrix of the waveplate may be representedby:

${T_{wp}( {\beta,{d\; \Delta \; n}} )} = {{R(\beta)}^{- 1}\begin{pmatrix}1 & 0 \\0 & e^{{- {jk}_{0}}d\; \Delta \; n}\end{pmatrix}{R(\beta)}}$

Because performance is sensitive to the angle of incidence, inaccordance with some embodiments of the permittivity is increased tobend the wave towards the normal direction as it propagates through thestructure in accordance with Snell's law. The angle of incidence isfurther increased by controlling the permittivity of the waveplates 102₁ . . . 102 _(N) in the x, y and z directions to reduce the indexcontrast between the two eigenpolarizations at oblique angles, whichcompensates for the increased optical thickness attributable to theimpedance matching layers 104 ₁, 104 ₂. For example, if the permittivityin the z-direction is increased such that

${ɛ_{1} = \begin{pmatrix}1.02 & 0.05 & 0 \\0.05 & 1.16 & 0 \\0 & 0 & 1.08\end{pmatrix}},{ɛ_{2} = \begin{pmatrix}1.08 & 0.02 & 0 \\0.02 & 1.01 & 0 \\0 & 0 & 1.04\end{pmatrix}},$

the transmission coefficient and axial ratio at 45° scan in the E and Hplanes as shown in graphical representation of FIG. 2. The response atoblique angles is analogous to a broadside case when the z-directedpermittivity is properly chosen. The z-directed permittivity may be nearthe geometric mean of the u and v directed permittivities (i.e., thetransverse permittivities along the principal axes) for the polarizer tobe operable at wide scan angles.

Referring further to FIGS. 1A and 1B, each dielectric matching assemblyof impedance matching layers 104 ₁, 104 ₂ impedance matches theartificial dielectrics to free space. Each dielectric assembly 104 ₁,104 ₂ includes a first section 106 ₁ having a first permittivity ∈₁, asecond section 106 ₂ having a second permittivity ∈₂ greater than thefirst permittivity, and a third section 106 ₃ having a thirdpermittivity ∈₃ greater than the second permittivity. These sections maybe referred to as “matching layers.” The thickness and orientation ofthe different sections can be optimized using a genetic algorithm aswill be appreciated by those skilled in the art. An exemplary expedientwas implemented in MATLAB®, where for simplicity, the anisotropicdielectric slabs all have the same permittivity. In total there are 14degrees of freedom that need to be optimized: thickness and permittivityof the 3 matching layers (6 unknowns), and thickness and orientation ofthe 4 anisotropic dielectrics (8 unknowns).

The cost function that is minimized is given by,

$= {\sum\limits_{\omega}\lbrack {( {1 + {{T_{Lx}( {\omega,{0{^\circ}},{0{^\circ}}} )}} - {{T_{Rx}( {\omega,{0{^\circ}},{0{^\circ}}} )}}} )^{5} + {\sum\limits_{\varphi}( \frac{( {1 + {{T_{Lx}( {\omega,{60{^\circ}},\varphi} )}} - {{T_{Rx}( {\omega,{60{^\circ}},\varphi} )}}} )^{5}}{10} )}} \rbrack}$

where T_(Rx)(ω, θ, ϕ) and T_(Lx)(ω, θ, ϕ) are the transmissioncoefficients when excited with a plane wave at a given frequency andangle of incidence.

This cost function maximizes T_(Rx) and minimizes T_(Lx) which minimizesinsertion loss and axial ratio over the desired bandwidth and angles ofincidence. The transmission coefficients are calculated at 21 frequencypoints between approximately 15 GHz and 70 GHz, and at angles ofincidence ϕ=0°, 60° and ϕ=−45°, 0°, 45°, 60°. A larger weight isassigned to the transmission coefficients at normal incidence. Thesummed elements within the cost function (1+|T_(Lx)|−|T_(Rx)|) areraised to the 5th power, which helps optimize for the worst-casescenario. It should be emphasized that the cost function can beevaluated analytically (i.e. full wave simulations are not required),which leads to relatively quick convergence. The optimization processtakes on the order of 30 minutes to complete with a 24 core CPU runningat 2.5 GHz.

Once the optimal material permittivities and thicknesses are determined,each layer is physically implemented. The impedance matching layers arephysically realized by stacking together different substrates. Withreference again to FIGS. 1A and 1B, the effective permittivities of theimpedance matching layers are approximately 1.3, 1.8, and 2.2, withthicknesses equal to approximately 1.8 mm, 1.2 mm, and 1.6 mm,respectively. A broadband impedance match between free space and thecascaded waveplates is realized by gradually transitioning thepermittivity. The permittivity of the outermost dielectric is reducedfrom approximately 1.8 to 1.3, by milling trenches in the substrate.

A unit cell of the cascaded, anisotropic waveplates is shown in FIG. 3.It is designed using standard dielectric mixing formulas to realize theoptimized anisotropic permittivity. The principal axes of the unit cellare oriented along the u, v, and z directions. Each cell consists ofapproximately 0.5 mm thickness Rogers 4003 substrate (ε=3.55) patternedwith a copper patch that is approximately 0.1 mm×0.45 mm in size. Thecopper rectangle primarily increases the permittivity in the vdirection, while minimally affecting the permittivity in the u and zdirections. The small unit cell size reduces the effects of bothtemporal and spatial dispersion. The effective permittivity tensor ofthe unit cell was extracted by illuminating a 10-unit cell thick slabwith normally incident plane waves propagating in the z and udirections,

$\begin{pmatrix}ɛ_{uu} & ɛ_{uv} & ɛ_{uz} \\ɛ_{vu} & ɛ_{vv} & ɛ_{vz} \\ɛ_{zu} & ɛ_{zv} & ɛ_{zz}\end{pmatrix} = \begin{pmatrix}2.45 & 0 & 0 \\0 & 4.1 & 0 \\0 & 0 & 3.2\end{pmatrix}$

The orientation of the different layers are β=9°, β=34°, β=29°, andβ=87°, for the first through fourth layers, respectively. The thickness(length) of the respective layers is approximately t₁=7.75 mm, t₂=3.25mm, t₃=4.25 mm, and t₄=4.00 mm.

It will be understood by those skilled in the art that by increasing theanisotropy of the waveplate, the thickness can be reduced. In addition,this increases robustness to fabrication tolerances since theperformance of a waveplate is proportional to the difference in theindices of refraction along the principal directions (i.e., ε_(vv)−√{square root over (ε_(uu))}). For example, a single waveplateilluminated at normal incidence with ε_(uu)=3.2 and ε_(vv)=3.5 convertsan incident linear polarization to circular polarization. If thepermittivity of ε_(vv)=3.5 is reduced by approximately 5% due tomanufacturing tolerances, the axial ratio of the transmitted field willincrease from approximately 0 dB to 7.5 dB. However, if the designedpermittivity contrast is increased such that ε_(uu)=2 and ε_(vv)=3.5,then a 5% decrease in ∈_(vv) only increases the axial ratio to 1 dB. Atthe same time, the permittivity contrast should not be increased morethan approximately 15% since this makes it more difficult to impedancematch the waveplates to free space using isotropic dielectrics.

The cascaded waveplates typically cannot be simulated as a single unitcell in a periodic lattice since the principal axes of the anisotropiclayers are all different. Therefore, the simulated S-parameters of thepolarizer are typically calculated by cascading the S-parameters of theindividual waveplates. This technique assumes the field at the boundarybetween two different waveplates is accurately represented by thefundamental Floquet modes, which are propagating plane waves with TE andTM polarizations. In other words, the simulation neglects evanescentcoupling between the different waveplates, which is expected tocontribute only minor influences on the polarizer's response. Note thatthe circuit solver in the HFSS® modeling tool provides a convenientmethod of cascading the S-parameters of the individual waveplates.

Referring now to FIGS. 4A, 4B and 4C, there is depicted the simulatedtransmission coefficient and axial ratio when illuminated with anx-polarized plane wave for different angles of incidence. At normalincidence, the transmission coefficient (T_(Rx)) is above −1 dB betweenapproximately 11 GHz and 72 GHz, and the axial ratio (AR) is below 3 dBfrom approximately 15 GHz to 70 GHz (4.7:1 bandwidth). This polarizeralso performs well at oblique incidence as shown in FIGS. 4B and 4C.

In accordance with an embodiment of the invention, an ultra-widebandlinear-to-circular polarizer 100 is realized by modifying theconventional geometry of a meanderline polarizer. As described above, byrotating the principal axes of the various layers it is possible toincrease the operable degrees-of-freedom, which can be leveraged toenhance bandwidth. Therefore, the orientation of each sheet is a freevariable that is optimized. Furthermore, each sheet is not restricted toonly meanderline geometries, which provides additional degrees offreedom. In other words, the layers are best represented as general,anisotropic sheet impedances.

A section of an exemplary cascaded sheet impedance polarizer is depictedin FIGS. 5A, 5B, 5C and 5D. Algorithm optimization may be utilized todesign the polarizer. The polarizer includes impedance matching layers104 ₁, 104 ₂ on the outside, and cascaded anisotropic metallic patternsprinted on Rogers 4003 substrates (sheets 102 ₁, 102 ₂, . . . 102 ₈)disposed between impedance matching layers 104 ₁, 104 ₂ and rotatedrelative to the z-axis as described above. In an example, thepermittivity of the 4003 substrate (ε=3.55) is large enough to improvethe performance at wide scan angles, but not too large to enable abroadband impedance-match to free space. In total, 8 patterned coppersheets are used, which are spaced approximately 0.4 mm apart in thez-direction. This results in roughly 25 unknowns that need to beoptimized: thickness and permittivity of the 3 matching layers (6unknowns), and dimensions (roughly 11 unknowns) and orientation (8unknowns) of the patterned metallic sheets. Again, the cost functiondescribed above is utilized to minimize insertion loss and axial ratioover the operational bandwidth at normal and oblique angles ofincidence.

Two different metallic geometries are considered for each sheet:meanderline and metallic patches, as shown FIGS. 5C and 5D. In thisimplementation, parametric sweeps were performed using ANSYS HFSS® toextract the anisotropic sheet impedances of the patterned metallicgeometries as a function of their dimensions (L_(m), P_(u), and L_(p))and frequency, at normal incidence. Simulations demonstrated that thesheet impedance is not a strong function of the angle of incidence.Interpolation may be utilized to approximate the sheet impedance ofgeometries that are not explicitly simulated. The dimension L_(m)primarily controls the inductance of the meanderline in the v-direction,while P_(u) determines the capacitance in the u-direction. The dimensionL_(p) primarily affects the capacitance of the patch along thev-direction. Again, the u and v directions correspond to the principalaxes of each sheet, which are rotated by an angle β relative to theglobal xy coordinate system. The simulated sheet impedances are insertedinto a MATLAB® routine that analytically calculates the S-parameters ofthe cascaded structure and the dimensions and orientation of each sheetmay optimized.

A brute force sweep may be used to determine which sheets utilizemeanderline geometries and which sheets utilize patches. First, everysheet is forced to be of the metallic patch geometry, and the algorithmfinds the minimum cost for this case by optimizing L_(p), and β of eachsheet, as well as the permittivity and thickness of the impedancematching layers. Then, the first sheet is replaced with the meanderlinegeometry and again the minimum cost is calculated using the geneticalgorithm. This process is repeated until every possible combination ofmeanderline and patch geometry is considered, of which there are a totalof 2⁸=256 combinations. At the end, the meanderline/patch combinationwith the lowest calculated cost is chosen. The optimal combinationutilizes meanderline geometries on the first, third, and seventh sheets.However, other options may be utilized to provide similar performance,with this implementation being merely exemplary.

The optimized dimensions of each patterned metallic sheet are shown inthe following table:

Sheet# L_(m) (mm) P_(u) (mm) L_(p) (mm) □ (deg.) 1 0.28 0.98 NA 5 2 NANA 0.63 118 3 0.60 0.84 NA 46 4 NA NA 0.70 143 5 NA NA 0.55 126 6 NA NA0.78 119 7 0.60 1.10 NA 89 8 NA NA 0.76 60

The effective permittivities of the impedance matching layers shown inFIG. 5A are approximately 1.3, 1.8, and 3.0, with thicknesses equal toapproximately 1.8 mm, 1.2 mm, and 0.75 mm, respectively.

Since it may be inefficient to rigorously simulate the entire polarizerusing a full-wave solver, the S-parameters of the different layers arecascaded together using the circuit solver in the HFSS® modeling tool tocalculate the S-parameters of the overall structure. Full wavesimulations of similar geometries that are periodic verified that simplycascading S-parameters provides an accurate estimate of the overallperformance. In other words, evanescent coupling between the differentlayers can be neglected for these cells sizes and interlayer spacing.The simulated performance is shown graphically in FIGS. 6A, 6B and 6C.

At normal incidence, the transmission coefficient (T_(Rx)) is aboveapproximately −1 dB between approximately 15 GHz and 72 GHz, and theaxial ratio is below approximately 3 dB from approximately 16 GHz to 68GHz (4.2:1 bandwidth). When illuminated at 60° from normal incidence inthe E, H, and diagonal planes, the peak axial ratio increases toapproximately 4 dB within the operating band. In this regard, thepolarizer performs well at oblique angles of incidence.

Linear-to-circular polarizers in accordance with embodiments of theinvention may be fabricated and measured using a Gaussian beamtelescope. In an exemplary embodiment, this system generates an incidentGaussian beam with beam waist diameter roughly equal to 3λ, whichsignificantly reduces the required fabricated area compared to the casewhere a single lens or no lenses are used. The system operates betweenapproximately 15 GHz and 110 GHz. The Gaussian beam telescope consistsof 2 linearly polarized standard gain horn antennas on either side ofthe polarizer under test. The horns have a high gain (˜23 dB), and theirradiated beams are quasi-Gaussian (85% coupling to the fundamentalGaussian mode). In order to characterize the polarizers across the wideoperating bandwidth, four different standard gain horn antennas wereused to cover the K, Ka, V, and W bands. The horns are connected to a2-port network analyzer that is integrated with frequency extenders toallow for measurements of the S-parameters up to 110 GHz. The systemutilizes 4 plano-convex Teflon® lenses with approximately 100 mmdiameters and approximately 150 mm focal lengths. The lenses areseparated from each other by the sum of their focal lengths (300 mm),which generates a collimated quasi-Gaussian beam at the center of thesystem with unity magnification at all operating frequencies. Thepolarizers are mounted on a 3D printed rotation stage that allows formeasuring the transmission coefficients at normal incidence and obliqueincidence, along different planes (e.g., E, H, and diagonal planes). Thebeam waist diameter at the lower operating frequencies (approximately 15GHz) is calculated to be ˜50 mm, and it reduces as the frequencyincreases. Therefore, the cross-sectional diameter of the polarizer inthis example needs to be approximately at least 50 mm. Orienting thepolarizer for measurements at oblique angles reduces the effectivecross-sectional area seen by the incident Gaussian beam. For example, a60° scan angle effectively reduces the polarizer's area by approximatelyone-half.

Linearly polarized horn antennas may be used to measure the polarizers.However, when characterizing the linear-to-circular transmission matrixit is helpful to have knowledge of the transmitted field along twoindependent polarizations. Conceptually, the simplest method ofcharacterizing the transmitted field is to first orient the receive hornto receive x-polarization, and then rotate the horn by 90° to receivey-polarization. Once T_(xx) and T_(yx) are known, it is straightforwardto calculate T_(Rx), T_(Lx), or equivalently, the transmitted axialratio. This approach may be less than desirable since the phase centerof the receive horn can easily shift when physically rotated. Thus, itis advantageous to first orient the two horns to measure T_(xx). Tomeasure an additional component of the transmitted polarization, awire-grid polarizer oriented along the x+y direction is inserted intothe path of the Gaussian beam, after the polarizer under test. Thetransmission coefficients of the wire-grid polarizer along its twoprincipal axes are independently measured so that its presence can beproperly calibrated. By utilizing measurements with and without thewire-grid polarizer in the beam's path, it is possible to extract thetransmitted field along two independent polarizations. Thesemeasurements are used to characterize T_(Rx) and the transmitted axialratio.

With reference to FIGS. 7A, 7B and 7C, a cascaded waveplate polarizer100 in accordance with an embodiment of the invention may be fabricatedby stacking together chemically etched printed-circuit-boards (PCBs)710. One of the fabricated PCBs 712 for the first waveplate 102 ₁ isdepicted in FIG. 7A. This PCB is stacked together with 83 identicalPCB's in the u-direction (see the unit cell illustrated in FIG. 3) toconstruct waveplate 102 ₁. A designed approximately 0.1 mm air gapbetween the stacked PCBs (in the u-direction) is realized by placingapproximately 0.1 mm shims between the boards at the edges thereof. Thesame process is employed to construct the other waveplates 102 ₂, 102 ₃,102 ₄. The 4 different cascaded waveplates are depicted in FIG. 7B. Inthe figure, the components are spaced apart for clarity. As shown inFIG. 7C, 3D printed holders 712 support each PCB in the properorientation.

With reference to FIG. 8, there is shown a graphical representation ofthe cascaded waveplate polarizer's measured and simulated transmissioncoefficient (T_(Rx)), and axial ratio (AR) at normal incidence. There isgood agreement between measurement and simulation. The gap in measuredfrequencies at 67 GHz corresponds to the location where the networkanalyzer switches modes between using internal signal generators (below67 GHz) and external frequency extenders (above 67 GHz). The measuredaxial ratio is below approximately 3 dB between approximately 19 GHz and76 GHz (4:1 bandwidth), and the insertion loss is below 2.5 dB over thisfrequency range. The insertion loss of the measurements is about 0.5 dBlarger than simulations, which could be due to fabrication tolerances,metal surface roughness, and/or air gaps between the 4 differentwaveplate sections and impedance matching layers.

With reference now to FIGS. 9A and 9B, a cascaded sheet impedancepolarizer is fabricated using standard PCB processing techniques. Eachsheet is chemically etched and then bonded together. The fabricatedsheets before bonding are shown in FIG. 9A, and the finished polarizerassembly after bonding and securing the impedance matching layers isdepicted in FIG. 9B.

With reference to FIGS. 10A, 10B and 10C, the measured and simulatedperformance of the cascaded sheet impedance polarizer is depicted forvarious angles of incidence. FIG. 10A shows a normal incidence. Themeasured insertion loss is below approximately 1.5 dB betweenapproximately 16 GHz and 73 GHz, and the axial ratio is belowapproximately 4.5 dB over this frequency range. There is reasonableagreement between simulation and measurement. The larger measured axialratio at 60 GHz may be attributed to fabrication tolerances. Thecascaded sheet impedance polarizer was also characterized at obliqueangles of incidence (θ=45°, 60°), along the E, H, and diagonal planes(ϕ=0°, 45°, 90°, 135°, 180°, 225°, 270°, 315° planes). The performanceonly slightly degrades as the angle of incidence is increased to 45°from normal, as shown in FIG. 10B. At an angle of incidence of 60° fromnormal, the average insertion loss and axial ratio increases by roughly1 dB compared to the broadside case as shown in FIG. 10C.

With reference now to FIG. 11, there is depicted an improvedlinear-to-circular polarizer in accordance with the embodiments of theinvention that may be realized by adding an additional anisotropic layerto compensate for the higher axial ratio near 60 GHz. A 3D printeddielectric grating made from VeroWhite (ε=2.8) is embedded within theimpedance matching layers (sandwiched between the ε=3 and ε=1.8 layers).A side view of the designed grating is shown in the inset. The gratinggenerates a simulated relative phase shift difference of approximately7° between the u and v polarizations at 60 GHz, which in turn brings thetransmitted phase shift difference between x and y polarizations closerto the ideal 90° at the higher operating frequencies. The phase shiftdifference between u and v polarizations is directly proportional to thefrequency. Therefore, the additional grating negligibly affects theperformance at the lower operating frequencies. The measured performanceat normal incidence and 60° scan angles is shown in FIGS. 12A and 12B.The presence of the dielectric grating keeps the axial ratio belowapproximately 3 dB from approximately 17 GHz to 66 GHz at normal anglesof incidence (FIG. 12A). The polarizer's performance only marginallydegrades when illuminated at wide scan angles such as shown in FIG. 12B.

It will be appreciated that the devices and methods of fabricationdisclosed in accordance with embodiments of the invention are set forthby way of example and not of limitation. Absent an explicit indicationto the contrary, the disclosed devices, systems, and method steps may bemodified, supplemented, omitted, and/or re-ordered without departingfrom the scope of this invention. Numerous variations, additions,omissions, and other modifications will be apparent to one of ordinaryskill in the art. In addition, the order or presentation of method stepsin the description and drawings above is not intended to require thisorder of performing the recited steps unless a particular order isexpressly required or otherwise clear from the context.

It will be understood by those skilled in the art that various changesmay be made in the form and details of the described embodimentsresulting in equivalent embodiments that remain within the scope of theappended claims.

What is claimed is:
 1. A linear-to-circular polarizer, comprising: aplurality of cascaded waveplates having biaxial permittivity, eachcascaded waveplate having a principal axis rotated at different anglesrelative to an adjacent section about a z-axis of a 3-dimensional x, y,z coordinate system, each of the plurality of cascaded waveplatescomprising an assembly of printed circuit boards and a unit cell of anartificial anisotropic dielectric, and impedance matching layersdisposed adjacent the cascaded waveplates, the impedance matching layerscomprising a first assembly of impedance matching layers and a secondassembly of impedance matching layers, each of the first and secondassemblies of impedance matching layers comprising a first sectionhaving a first permittivity ∈₁, a second section having a secondpermittivity ∈₂ greater than the first emissivity, and a third sectionhaving a third permittivity ∈₃ greater than the second permittivity. 2.A linear-to-circular polarizer, comprising: a plurality of cascadedanisotropic sheets, each sheet having a principal axis rotated atdifferent angles relative to an adjacent sheet about a z-axis of a3-dimensional x, y, z coordinate system; and impedance matching layersdisposed adjacent the cascaded sheets.
 3. The linear-to-circularpolarizer of claim 2, where the cascaded sheets comprise anisotropicmetallic patterns.
 4. The linear-to-circular polarizer of claim 3, wherethe anisotropic metallic patterns have meanderline and metallic patchgeometries.
 5. The linear-to-circular polarizer of claim 2, where theimpedance matching layers comprise a first assembly of impedancematching layers and a second assembly of impedance matching layers, eachof the first and second assemblies of impedance matching layerscomprising a first section having a first permittivity ∈₁, a secondsection having a second permittivity ∈₂ greater than the firstpermittivity, and a third section having a third permittivity ∈₃ greaterthan the second permittivity.
 6. The linear-to-circular polarizer ofclaim 5, where each assembly of impedance matching layers comprises aplurality of different substrates.
 7. A linear-to-circular polarizer,comprising: a plurality of cascaded anisotropic sheets, each sheethaving a principal axis rotated at different angles relative to anadjacent sheet about a z-axis of a 3-dimensional x, y, z coordinatesystem, the cascaded sheets comprising anisotropic metallic patternshaving meanderline and metallic patch geometries; and impedance matchinglayers disposed adjacent the cascaded sheets, the impedance matchinglayers comprising a first assembly of impedance matching layers and asecond assembly of impedance matching layers, each of the first andsecond assemblies of impedance matching layers comprising a firstsection having a first permittivity ∈₁, a second section having a secondpermittivity ∈₂ greater than the first permittivity, and a third sectionhaving a third permittivity ∈₃ greater than the second permittivity. 8.The linear-to-circular polarizer of claim 2, further comprising a threedimensional (3D) printed dielectric grating embedded between theimpedance matching layers.